Suppose the joint distribution of x and y is symmetric

Question

Question: Symmetry under rotations.

a) Suppose the joint distribution of X and Y is symmetric under rotations, Are X and Y necessarily independent? Are they necessarily uncorrelated? Explain by arguments or examples.

b) Suppose (X, Y) is a point picked at random from the unit circle X² + Y² = 1, Calculate E(X²), E(Y²), and E(XY).

c) Suppose U is uniformly distributed on (0, 1), X = Cos 2πU, Y = Sin 2πU. Are X and Y uncorrelated? Are X and Y independent? Explain carefully the connection between b) and c).